On the structure of Weyl modules

Abstract

Let M x (resp. Vx) denote the irreducible module (resp. the Weyl module) with highest weight Z of a connected reductive algebraic group G over k, with k an algebraically closed field of characteristic p >0 . In [3] we proved that a necessary condition for M~ to be a composition factor of V X is that 2 is strongly linked to )6 We proved more generally that the same is true for all cohomology groups of the line bundles on G/B, B a Borel subgroup, induced by )~ and its conjugates under the Weyl group W. Even though Very little is known about the higher cohomology groups of line bundles on G/B (in particular it is not known exactly when they are indecomposable as G-modules) the existence of certain canonical maps between them enables us to deduce some properties of their composition factors and of the structure of Weyl modules. In this paper we shall explore the method from [3] somewhat further. First we obtain an estimate on the set of weights in HI(D, i>0, ;g a character o r b (as in [3] we write Hi00 short for Hi(G/B, L00)). We prove that if )~ is dominant then Z is the unique highest weight in H~(W)(w.z) for all w~W, and it occurs with multiplicity 1. Here ~(w) denotes the length of w and so in characteristic zero, Ht('~)(w.z) is the unique non-vanishing cohomology group [7]. Next we prove that when 2 and X are dominant and 2 is maximal among the weights strongly linked to ;g then M~ occurs exactly once as a composition factor of V z. For p-regular weights this result was proved by Jantzen [9, Satz 10] (see also Carter's and Lusztig's results for groups of type A, [5, Theorem 4.1]). The above result on the highest weight in Ht(W)(w.x) gives a canonical homomorphism Ht{W)(w.z)-~H~ In this way we obtain for each reduced expression for w0, the longest element in W, a filtration of H~ (and of V.~). Using these filtrations we are able to prove the conjecture of Carter and Lusztig [5, p. 239] about intertwining homomorphisms between certain Weyl modules in most cases. We conclude the paper with a general formula for the character of the image of intertwining homomorphisms in the direction of a simple root. As illustrations we give details in the 2 simplest non-trivial cases: type A 2 and B2, Z in the bottom p<alcove.